Transfer Matrix Description Research Articles

Layered piezoelectric structures with arbitrary acoustic termination impedances.

Multilayer piezoelectric transducers and resonators are widely used for generating propagating and standing acoustic waves as well as for sensor devices. More recently, layered piezoelectric structures based on thin film technology became increasingly important for electromechanical filters used in mobile phones. As a consequence, analytical mathematical modeling of such structures is of high interest. In this paper, a general rigorous transfer matrix description for one-dimensional (1D) layered structures consisting of piezoelectric, visco-elastic, and dielectric layers of arbitrary number is introduced to characterize the electrical and mechanical behavior of a general piezoelectric transducer or resonator with two electrodes and arbitrary acoustic termination impedances (Rig-1D-model). This model is the most general 1D analytical description of layered piezoelectric structures available and can be used for the characterization of various composite transducer and resonant sensor applications. Considered in detail are layered structures with the technically important cases of only one electromechanically coupled mode, and the structure at one or both outer surfaces is in contact with semi-infinite media. For such devices, it is shown how the frequency dependence of the total electrical admittance and spatial dependence of the displacements can be calculated.

Tracking the trajectory of an object in a noisy environment with unknown statistics: A novel robust Kalman filter residue-based approach

A novel robust Kalman filter (KF)-based controller design approach is proposed to accurately track a specified trajectory under unknown stochastic disturbance, known deterministic disturbance, and measurement noise. The system is nonlinear and is approximated by a piecewise-linear dynamic model. The Box–Jenkins model is an augmented model of the signal and the disturbance, is non-controllable and observable, while the signal model is controllable and observable. An emulator-based two-stage identification is employed to obtain an accurate system model needed to design the robust controller. The system and its KF are identified, and the signal and output errors are estimated. From the identified models, the signal, its KF, the disturbance model, and the whitening filter are all obtained using balanced model reduction techniques. It is shown that the signal model is a transfer matrix description relating the system output and KF residual, and the residual is the whitened output error. The disturbance model is identified by inverse filtering. A new combined feedforward–feedback controller is designed and implemented using an internal model of the reference driven by both the error between the reference and the signal estimate, and by the feedforwarded reference signal. The proposed scheme was successfully evaluated on a simulated autonomously guided drone.

Solvable non-Hermitian skin effect in many-body unitary dynamics

We study unitary evolution of bipartite entanglement in a circuit with nearest-neighbor random gates. Deriving a compact non-unitary description of purity dynamics on qudits we find a sudden transition in the purity relaxation rate the origin of which is in the underlying boundary localized eigenmodes -- the skin effect. We provide the full solution of the problem, being one of the simplest iterations of two-site matrices, namely, that each is a sum of only two projectors. This leads to rich dynamics influenced by the Jordan normal form of the kernel and, most importantly, a spectrum that is completely discontinuous in the thermodynamic limit. It provides a simple example of how a seemingly innocuous many-body unitary evolution can harbor interesting mathematical effects: an effective non-symmetric Toeplitz transfer matrix description causes a phantom relaxation, such that the correct relaxation rate is not given by the matrix spectrum, but rather by its pseudospectrum.

Dynamics around the site percolation threshold on high-dimensional hypercubic lattices.

Recent advances on the glass problem motivate reexamining classical models of percolation. Here we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical dimension of simple percolation, d_{u}=6. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for d≥d_{u}. The theoretical derivation, which considers Bethe lattices with generalized connectivity and a random graph model, confirms that logarithmic scalings should persist in the limit d→∞. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_{u} as well as their logarithmic scaling above d_{u}. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.

Exact impact response of multi-layered cement-based piezoelectric composite considering electrode effect

Multi-layered cement-based piezoelectric composites could enable accurate real-time detection of the concrete structure deformation induced by impact load. An analytical method for quantifying the impact response of the multi-layered cement-based piezoelectric composite is established based on the piezo-elasticity, and a general transfer matrix description for the composite with any number of layers is derived. The motion of the composite is decomposed into natural modes according to its physical significance of vibration modes. The mechanical and electrical solutions are obtained via the mode summation method and the virtual work principle. In order to give a clear demonstration, some numerical simulations are conducted to verify the validity of the theoretical analysis. Moreover, the current analytical method considers the electrode as an extra layer and evaluates the effect of its thickness and material on the performance of the multi-layered cement-based piezoelectric composite. It can be seen that the mathematical model presented in this article provides a rigorous tool for the analysis of the multi-layered cement-based piezoelectric composite and therefore could benefit the design of certain types of smart devices under impact load.

Application of transfer matrix and transfer function analysis to grating-type dielectric laser accelerators: Ponderomotive focusing of electrons

The question of suitability of transfer matrix description of electrons traversing grating-type dielectric laser acceleration (DLA) structures is addressed. It is shown that although matrix considerations lead to interesting insights, the basic transfer properties of DLA cells cannot be described by a matrix. A more general notion of a transfer function is shown to be a simple and useful tool for formulating problems of particle dynamics in DLA. As an example, a focusing structure is proposed which works simultaneously for all electron phases.

Multivariable filtered Smith predictor for systems with sinusoidal disturbances

ABSTRACTThis paper presents a new design condition for the multivariable filtered Smith predictor (MFSP) in order to achieve sinusoidal disturbance rejection in steady state. The proposed MFSP strategy is based on a transfer matrix description with multiple delays. A suitable prediction error filter is presented to deal with this new design condition. The proposed design requirement can be used in model predictive controllers with inactive constraints in steady state. Two case studies are used to illustrate the main benefits of the proposed framework.

Integrability and conformal data of the dimer model

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a description. Using Lieb’s transfer matrix and its description in terms of the Temperley–Lieb algebra at , we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analyzed in the scaling limit, and the result for is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of and are found to yield a realization of the Virasoro algebra, familiar from fermionic bc ghost systems. In this realization, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin–Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the conformal integrals of motion. Consistent with the expression for obtained from the transfer matrix, we also construct higher Virasoro modes with c = 1 and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the c = 1 integrals of motion. Although this indicates that Lieb’s transfer matrix description is incompatible with the c = 1 interpretation, it does not rule out the existence of an alternative, c = 1 compatible, transfer matrix description of the dimer model.

Analytical Sensor Response Function of Viscosity Sensors Based on Layered Piezoelectric Thickness Shear Resonators

Resonant piezoelectric sensors based on bulk acoustic wave (BAW) thickness shear resonators are promising for the inline measurement of fluid viscosity, e.g., in industrial processes. The sensor response function can be derived from the general rigorous transfer matrix description of one-dimensional layered structures consisting of piezoelectric and non-piezoelectric layers of arbitrary number. This model according to Nowotny et al. provides a complete analytical description of the electrical and mechanical behaviour of such structures with two electrodes and arbitrary acoustic termination impedances (Rig-1d-Model). We apply this model to derive the sensor response functions and the mechanical displacement curves of the following configurations appropriate for viscosity sensors:(a)An AT cut quartz crystal plate in contact with vacuum at the backside plane and with the liquid under investigation at the front side plane (QL).(b)An AT cut quartz crystal in contact with the liquid under investigation at both sides (LQL).It is shown that in the QL case the originally only heuristically introduced and well established sensor response function according to Kanasawa can be derived from the Rig-1d-Model by introducing minor approximations. Experimental results are presented for the LQL configuration using an N1000 viscosity reference oil as test fluid.

Symmetry Properties of Electromagnetic Planar Arrays in Transfer Matrix Description

We analyze in this paper symmetry properties of planar arrays in the long-wave approximation and for general case of oblique incidence of electromagnetic waves. We introduce the concept of the group of wave vectors for scattering problems in transfer matrix description. Using formalism of magnetic groups we define all the possible symmetries for planar arrays and show how to calculate the structure of 4 × 4 transfer matrix for nonmagnetic and magnetic arrays. We discuss also the reciprocity conditions for the transfer matrices which are different from those in scattering matrix description. Application of the theory is illustrated by some examples.

Multicritical point of Ising spin glasses on triangular and honeycomb lattices

The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated, with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-paramagnetic phase boundary is probed, with the following estimates for the associated critical indices: $\nu=1.49(2)$, $\gamma=2.71(4)$, $\eta_1= 0.183(3)$, distinctly different from the percolation values.

Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal

We present a modeling of a degenerate four-wave-mixing nonlinear process in one-dimensional photonic crystals. The model is based on the nonlinear extension of the transfer-matrix description of propagation in the structure. The influence of light localization, near the band edge of the structure, on the enhancement of the phase-conjugate reflectivity is studied. The phase-conjugate reflectivity is shown to increase as the eighth power of the number of layers with an additional large dependence on the index contrast of the structure. In both cases the enhancement is accompanied by a strong reduction of the resonance width, which may lead to some limitation of the enhancement when ultrashort pulses are used. A strong influence of the losses on the nonlinear efficiency of the structure is also predicted with a great importance of scattering losses at the multiple interfaces of the structure.

Simultaneous measurement of middle-ear input impedance and forward/reverse transmission in cat

Reported here is a technique for measuring forward and reverse middle-ear transmission that exploits distortion-product otoacoustic emissions (DPOAEs) to drive the middle ear "in reverse" without opening the inner ear. The technique allows measurement of DPOAEs, middle-ear input impedance, and forward and reverse middle-ear transfer functions in the same animal. Intermodulation distortion in the cochlea generates a DPOAE at frequency 2f1-f2 measurable in both ear-canal pressure and the velocity of the stapes. The forward transfer function is computed from stapes velocities and corresponding ear-canal pressures measured at the two primary frequencies; the reverse transfer function is computed from velocity and pressure measurements at the DPOAE frequency. Middle-ear input impedance is computed from ear-canal pressure measurements and the measured Thévenin equivalent of the sound-delivery system. The technique was applied to measure middle-ear characteristics in anesthetized cats with widely opened middle-ear cavities (0.2-10 kHz). Stapes velocity was measured at the incudo-stapedial joint. Results on five animals are reported and compared with a published middle-ear model. The measured forward transfer functions and input impedances generally agree with previous measurements, and all measurements agree qualitatively with model predictions. The reverse transfer function is shown to depend on the acoustic load in the ear canal, and the measurements are used to compute the round-trip middle-ear gain and delay. Finally, the measurements are used to estimate the parameters of a two-port transfer-matrix description of the cat middle ear.

Transfer-matrix description of heterostructures involving superconductors and ferromagnets

Based on the technique of quasiclassical Green's functions, we construct a theoretical framework for describing heterostructures consisting of superconductors and/or spin-polarized materials. The necessary boundary conditions at the interfaces separating different metals are formulated in terms of hopping amplitudes in a t-matrix approximation. The theory is applicable for an interface with arbitrary transmission and exhibiting scattering with arbitrary spin dependence. Also, it can be used in describing both ballistic and diffusive systems. We establish the connection between the standard scattering-matrix approach and the existing boundary conditions, and demonstrate the advantages offered by the t-matrix description.

Connectivity-dependent properties of diluted sytems in a transfer-matrix description

We introduce an approach to connectivity-dependent properties of diluted systems, which is based on the transfer-matrix formulation of the percolation problem. It simultaneously incorporates the connective properties reflected in nonzero matrix elements and allows one to use standard random-matrix multiplication techniques. Thus it is possible to investigate physical processes on the percolation structure with the high efficiency and precision characteristic of transfer-matrix methods, while avoiding disconnections. The method is illustrated for two-dimensional site percolation by calculating the finite-size longitudinal dc conductivity for (i) the critical correlation length along the strip, (ii) at the percolation threshold, and (iii) very near the pure-system limit.

Complex-energy analysis of intrinsic lifetimes of resonances in biased multiple quantum wells.

The electric-field-dependent intrinsic lifetimes of resonances in biased multiple qunatum wells are studied by using a complex-energy analysis based on an Airy-function transfer-matrix description of tunneling. Special attention is paid to the case when at some particular electric field two resonances belonging to different quantum wells resonantly align in energy. Two different characteristic behaviors were found in such electric-field sweeps: Long-lived resonances typically exhibit an anticrossing of their quasieigenenergies and a corresponding crossing of their lifetimes, usually associated with large changes in these lifetimes over many orders of magnitude, while short-lived resonances feature a crossing of their quasieigenenergies and an anticrossing of their lifetimes with less variation in these lifetimes. The parameters relevant for these two different regimes are discussed, and a simple model is derived to describe the crossover between these regimes.

Ground states of VBS models on cayley trees

We study the thermodynamic limit of the ground states of VBS models on a Cayley tree. We prove uniqueness for coordination numbersz ⩽ 4 and the occurrence of Neel order forz ⩾ 5. Our main technical tool is a transfer matrix description of VBS states.

Layered piezoelectric resonators with an arbitrary number of electrodes (general one-dimensional treatment)

A general transfer matrix description for one-dimensional layered structures consisting of piezoelectric and nonpiezoelectric anisotropic layers of arbitrary number is used to calculate the electrical admittance matrix for such resonators with N electrodes. The calculation is done in detail for linearly stacked resonators with two free surfaces as well as for ring resonators with a closed acoustical path. Experimental and theoretical results are given and compared for a ring resonator with two piezoelectric layers excited by four electrodes. Such a configuration can be used to generate unidirectional resonant waves.

Motional capacitance of layered piezoelectric thickness-mode resonators

The Butterworth-Van Dyke equivalent circuit for description of the electrical behavior of piezoelectric bulk resonators is considered. The motional capacitance, C(1), in the circuit characterizes the strength of piezoelectric excitability of a vibration mode. For layered one-dimensional (1-D) structures this parameter can be calculated from the admittance given by the transfer matrix description of H. Nowotny and E. Benes (1987). Introducing the equivalent area of a vibration mode, the calculation is generalized for the three-dimensional (3-D) case of thickness-mode vibration amplitudes varying only slowly in the lateral directions. Detailed formulae are given for the case of singly rotated quartz crystals or ultrasonic transducers with additional layers on one or two sides. Good agreement of the calculated C (1) with experimental data is shown for mass-loaded planoconvex AT-cut quartz crystals.

Ray transfer matrix analysis of multiple-sprism dye laser oscillators

A ray transfer matrix description of multiple-prism beam expanders used to produce narrow-linewidth emission in pulsed dye lasers is given. The formulae are then applied to describe beam divergence characteristics of a narrow-linewidth multiple-prism grating oscillator. A comparison between theory and measured beam divergence and laser linewidth is provided.